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In a previous post, I discussed some methods for showing that a given group contains a (nonabelian) free subgroup. The methods were analytic and/or dynamical, and phrased in terms of the existence (or nonexistence) of certain functions on $G$ or on spaces derived from $G$, or in terms of actions of $G$ on certain spaces. Dually, one can try to find a free group in $G$ by finding a homomorphism $\rho: F \to G$ and looking for circumstances under which $\rho$ is injective.

For concreteness, let $G = \pi_1(X)$ for some (given) space $X$. If $F$ is a free group, a representation $\rho:F \to G$ up to conjugation determines a homotopy class of map $f: S \to X$ where $S$ is a $K(F,1)$. The most natural $K(F,1)$‘s to consider are graphs and surfaces (with boundary). It is generally not easy to tell whether a map of a graph or a surface to a topological space is $\pi_1$-injective at the topological level, but might be easier if one can use some geometry.

Example: Let $X$ be a complete Riemannian manifold with sectional curvature bounded above by some negative constant $K < 0$. Convexity of the distance function in a negatively curved space means that given any map of a graph $f:\Gamma \to X$ one can flow $f$ by the negative gradient of total length until it undergoes some topology change (e.g. some edge shrinks to zero length) or it (asymptotically) achieves a local minimum (the adjective “asymptotically” here just means that the flow takes infinite time to reach the minimum, because the size of the gradient is small when the map is almost minimum; there are no analytic difficulties to overcome when taking the limit). A typical topological change might be some loop shrinking to a point, thereby certifying that a free summand of $\pi_1(\Gamma)$ mapped trivially to $G$ and should have been discarded. Technically, one probably wants to choose $\Gamma$ to be a trivalent graph, and when some interior edge collapses (so that four points come together) to let the $4$-valent vertex resolve itself into a pair of $3$-valent vertices in whichever of the three combinatorial possibilities is locally most efficient. The limiting graph, if nonempty, will be trivalent, with geodesic edges, and vertices at which the three edges are all (tangentially) coplanar and meet at angles of $2\pi/3$. Such a graph can be certified as $\pi_1$-injective provided the edges are sufficiently long (depending on the curvature $K$). After rescaling the metric on $X$ so that the supremum of the curvatures is $-1$, a trivalent geodesic graph with angles $2\pi/3$ at the vertices and edges at least $2\tanh^{-1}(1/2) = 1.0986\cdots$ is $\pi_1$-injective. To see this, lift to maps between universal covers, i.e. consider an equivariant map from a tree $\widetilde{\Gamma}$ to $\widetilde{X}$. Let $\ell$ be an embedded arc in $\widetilde{\Gamma}$, and consider the image in $\widetilde{X}$. Using Toponogov’s theorem, one can compare with a piecewise isometric map from $\ell$ to $\mathbb{H}^n$. The worst case is when all the edges are contained in a single $\mathbb{H}^2$, and all corners “bend” the same way. Providing the image does not bend as much as a horocircle, the endpoints of the image of $\ell$ stay far away in $\mathbb{H}^2$. An infinite sided convex polygon in $\mathbb{H}^2$ with all edges of length $2\tanh^{-1}(1/2)$ and all angles $2\pi/3$ osculates a horocycle, so we are done.

Remark: The fundamental group of a negatively curved manifold is word-hyperbolic, and therefore contains many nonabelian free groups, which may be certified by pingpong applied to the action of the group on its Gromov boundary. The point of the previous example is therefore to certify that a certain subgroup is free in terms of local geometric data, rather than global dynamical data (so to speak). Incidentally, I would not swear to the correctness of the constants above.

Example: A given free group is the fundamental group of a surface with boundary in many different ways (this difference is one of the reasons that a group like $\text{Out}(F_n)$ is so much more complicated than the mapping class group of a surface). Pick a realization $F = \pi_1(S)$. Then a homomorphism $\rho:F \to G$ up to conjugacy determines a homotopy class of map from $S$ to $X$ as above. If $X$ is negatively curved as before, each boundary loop is homotopic to a unique geodesic, and we may try to find a “good” map $f:S \to X$ with boundary on these geodesics. There are many possible classes of good maps to consider:

1. Fix a conformal structure on $S$ and pick a harmonic map in the homotopy class of $f$. Such a map exists since the target is nonpositively curved, by the famous theorem of Eells-Sampson. The image is real analytic if $X$ is, and is at least as negatively curved as the target, and therefore there is an a priori upper bound on the intrinsic curvature of the image; if the supremum of the curvature on $X$ is normalized to be $-1$, then the image surface is $\text{CAT}(-1)$, which just means that pointwise it is at least as negatively curved as hyperbolic space. By Gauss-Bonnet, one obtains an a priori bound on the area of the image of $S$ in terms of the Euler characteristic (which just depends on the rank of $F$). On the other hand, this map depends on a choice of marked conformal structure on $S$, and the space of such structures is noncompact.
2. Vary over all conformal structures on $S$ and choose a harmonic map of least energy (if one exists) or find a sequence of maps that undergo a “neck pinch” as a sequence of conformal structures on $S$ degenerates. Such a neck pinch exhibits a simple curve in $S$ that is essential in $S$ but whose image is inessential in $X$; such a curve can be compressed, and the topology of $S$ simplified. Since each compression increases $\chi$, after finitely many steps the process terminates, and one obtains the desired map. This is Schoen-Yau‘s method to construct a stable minimal surface representative of $S$. When the target is $3$-dimensional, the surface may be assumed to be unbranched, by a trick due to Osserman.
3. Following Thurston, pick an ideal triangulation of $S$ (i.e. a geodesic lamination of $S$ whose complementary regions are all ideal triangles); since $S$ has boundary, we may choose such a lamination by first picking a triangulation (in the ordinary sense) with all vertices on $\partial S$ and then “spinning” the vertices to infinity. Unless $\rho$ factors through a cyclic group, there is some choice of lamination so that the image of $f$ can be straightened along the lamination, and then the image spanned with $CAT(-1)$ ideal triangles to produce a pleated surface in $X$ representing $f$ (note: if $X$ has constant negative curvature, these ideal triangles can be taken to be totally geodesic). The space of pleated surfaces in fixed (closed) $X$ of given genus is compact, so this is a reasonable class of maps to work with.
4. If $G$ is merely a hyperbolic group, one can still construct pleated surfaces, not quite in $X$, but equivariantly in Mineyev’s flow space associated to $\widetilde{X}$. Here we are not really thinking of the triangles themselves, but the geodesic laminations they bound (which carry the same information).
5. If $X$ is complete and $3$-dimensional but noncompact, the space of pleated surfaces of given genus is generally not compact, and it is not always easy to find a pleated surface where you want it. This can sometimes be remedied by shrinkwrapping; one looks for a minimal/pleated/harmonic surface subject to the constraint that it cannot pass through some prescribed set of geodesics in $X$ (which act as “barriers” or “obstacles”, and force the resulting surface to end up roughly where one wants it to).

Anyway, one way or another, one can usually find a map of a surface, or a space of maps of surfaces, representing a given homomorphism, with some kind of a priori control of the geometry. Usually, this control is not enough to certify that a given map is $\pi_1$-injective, but sometimes it might be. For instance, a totally geodesic (immersed) surface in a complete manifold of constant negative curvature is always $\pi_1$-injective, and any surface whose extrinsic curvature is small enough will also be $\pi_1$-injective.

Geometric methods to certify injectivity of free or surface groups are very useful and flexible, as far as they go. Unfortunately, I know of very few topological methods to certify injectivity. By far the most important exception is the following:

Example: In $3$-dimensions, one should look for properly embedded surfaces. If $M$ is a $3$-manifold (possibly with boundary), and $S$ is a two-sided properly embedded surface, the famous Dehn’s Lemma (proved by Papakyriakopoulos) implies that either $S$ is $\pi_1$-injective, or there is an embedded essential loop in $S$ that bounds an embedded disk in $M$ on one side of $S$. Such a loop may be compressed (i.e. $S$ may be cut open along the loop, and two copies of the compressing disk sewn in) preserving the property of embeddedness, but increasing $\chi$. After finitely many steps, either $S$ compresses away entirely, or one obtains a $\pi_1$-injective surface. One way to ensure that $S$ does not compress away entirely is to start with a surface that is essential in (relative) homology; another way is to look for a surface dual to an action (of $\pi_1(M)$) on a tree. In the latter case, one can often construct quite different free subgroups in $\pi_1(M)$ by pingpong on the ends of the tree. Note by the way that this method produces closed surface subgroups as well as free subgroups. Note too that two-sidedness is essential to apply Dehn’s Lemma.

Remark: Modern $3$-manifold topologists are sometimes unreasonably indifferent to the power of Dehn’s Lemma (probably because this tool has been incorporated so fully into their subconscious?); it is worth reading Ralph Fox’s review of Papakyriakopoulos’s paper (linked above). Of this paper, he writes:

. . . it has already led to renewed attack on the problem of classifying the 3-dimensional manifolds; significant results have been and are being obtained. A complete solution has suddenly become a definite possibility.

Remember this was written more than 50 years ago — before the geometrization conjecture, before the JSJ decomposition, before the Scott core theorem, before Haken manifolds. The only reasonable reaction to this is: !!!

Example: The construction of injective surfaces by Dehn’s Lemma may be abstracted in the following way. Given a target space $X$, and a class of maps $\mathcal{F}$ of surfaces into $X$ (in some category; e.g. homotopy classes of maps, pleated surfaces, $\text{CAT}(-1)$ surfaces, etc.) suppose one can find a complexity $c:\mathcal{F} \to \mathcal{O}$ with values in some ordered set, such that if $f \in \mathcal{F}$ is not injective, one can find $f' \in \mathcal{F}$ of smaller complexity. Then if $\mathcal{O}$ is well-ordered, an injective surface may be found. If $\mathcal{O}$ is not well-ordered, one may ask at least that $c$ is upper semi-continuous on $\mathcal{F}$, and hope to extend it upper semi-continuously to some suitable compactification of $\mathcal{F}$. Even if $\mathcal{O}$ is not well-ordered, one can at least certify that a map is injective, by showing that it minimizes $c$. Here are some potential examples (none of them entirely satisfactory).

1. Given a (homologically trivial) homotopy class of loop $\gamma$ in $X$, one can look at all maps of orientable surfaces $S$ to $X$ with boundary factoring through $\gamma$. For such a surface, let $n(S)$ denote the degree with which the (possibly multiple) boundary (components) of $S$ wrap homologically around $\gamma$, and let $-\chi^-(S)$ denote the sum of Euler characteristics of non-disk and non-sphere components of $S$. For each surface $S$, one considers the quantity $-\chi^-(S)/2n(S)$ (the factor of $2$ can be ignored if desired). The important feature of this quantity is that it does not change if $S$ is replaced by a finite cover. If $\pi_1(S)$ is not injective, let $\alpha$ be an essential loop on $S$ whose image in $X$ is inessential. Peter Scott showed that any essential loop on a surface lifts to an embedded loop in some finite cover. Hence, after passing to such a cover, $\alpha$ may be compressed, and the resulting surface $S'$ satisfies $-\chi^-(S')/2n(S') < -\chi^-(S)/2n(S)$. In other words, a global minimizer of this quantity is injective. Such a surface is called extremal. The problem is that extremal surfaces do not always exist; but this construction motivates one to look for them.
2. Given a $\text{CAT}(-1)$ surface $S$ with geodesic boundary in $X$, one can retract $S$ to a geodesic spine, and encode the surface by the resulting fatgraph, with edges labelled by homotopy classes in $X$. Since Euler characteristic is local, one does not really care precisely how the pieces of the fatgraph are assembled, but only how many pieces of what kinds are needed for a given boundary. So if only finitely many such pieces appear in some infinite family of surfaces, one can in fact construct an extremal surface as above, which is necessarily injective (more technically, one reduces the computation of Euler characteristic to a linear programming problem, finds a rational extremal solution (which corresponds to a weighted sum of pieces of fatgraph), and glues together the pieces to construct the extremal surface; one situation in which this scheme can be made to work is explained in this paper of mine). Edges can be subdivided into a finite number of possibilities, so one just needs to ensure finiteness of the number of vertex types. One condition that ensures finiteness of vertex types is the existence of a uniform constant $C>0$ so that for each surface $S$ in the given family, and for each point $p \in S$, there is an estimate $\text{dist}(p,\partial S) \le C$. If this condition is violated, one finds pairs $p_i,S_i$ which converge in the geometric topology to a point in a complete (i.e. without boundary, but probably noncompact) surface.
3. Given $S \to X$, either compress an embedded essential loop, or realize $S$ by a least area surface. If $S$ is not injective, pass to a cover, compress a loop, and realize the result by a least area surface. Repeat this process. One obtains in this way a sequence of least area surfaces in $X$ (typically of bigger and bigger genus) and there is no reason to expect the process to terminate. If $X$ is a $3$-manifold, the curvature of a least area surface admits two-sided curvature bounds away from the boundary, by a theorem of Schoen (near the boundary, the negative curvature might blow up, but only in controlled ways — e.g. after rescaling about a sequence of points with the most negative curvature, one may obtain in the limit a helicoid). Away from the boundary, the family of surfaces one obtains vary precompactly in the $C^\infty$ topology, and one may obtain a complete locally least area lamination $\Lambda$ in the limit. If $\pi_1(\Lambda)$ is not injective, one can continue to pass to covers (applying a version of Scott’s theorem for infinite surfaces) and compress, and by transfinite induction, eventually arrive at a locally least area lamination with injective $\pi_1$. Of course, such a limit might well be a lamination by planes. However, the lamination one obtains is not completely arbitrary: since it is a limit of limits of . . . compact surfaces, one can choose a limit that admits a nontrivial invariant transverse measure (one must be careful here, since the lamination will typically have boundary). Or, as in bullet 2. above, one may insist that this limit lamination is complete (i.e. without boundary).

It is more tricky to find a limit lamination as in 3. without boundary and admitting an invariant transverse measure; in any case, this motivates the following:

Question: Is there a closed hyperbolic $3$-manifold $M$ which admits a locally least area transversely measured complete immersed lamination $\Lambda$, all of whose leaves are disks? (note that the answer is negative if one asks for the lamination to be embedded (there are several easy proofs of this fact)).

Secretly, the function that assigns $\inf_S -\chi^-(S)/2n(S)$ to a homologically trivial loop $\gamma$ is the stable commutator length of the conjugacy class in $\pi_1(X)$ represented by $\gamma$. Extremal surfaces can sometimes be certified by constructing certain functions on $\pi_1(X)$ called homogeneous quasimorphisms, but a discussion of such functions will have to wait for another post.